PRODUCT yield plays a critical role in determining the

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1 140 IEEE TRANSACTIONS ON SEMICONDUCTOR MANUFACTURING, VOL. 18, NO. 1, FEBRUARY 2005 Monitoring Defects in IC Fabrication Using a Hotelling T 2 Control Chart Lee-Ing Tong, Chung-Ho Wang, and Chih-Li Huang Abstract Monitoring the wafer defects in integrated circuit (IC) fabrication is essential for enhancing wafer yield. However, significant defect clustering occurs when the wafer is large, so the conventional defect control chart, based on the Poisson distribution, is inappropriate. Defect clustering must also be analyzed to monitor effectively defects in IC fabrication process control. This study developed a novel procedure using the multivariate Hotelling 2 control chart, based on the number of defects and the defect clustering index (CI) to monitor simultaneously the number of defects and the defect clusters. The CI does not require any statistical assumptions concerning the distribution of defects and can accurately evaluate the clustering phenomena. A case study of a Taiwanese IC manufacturer demonstrates the effectiveness of the proposed procedure. Index Terms C chart, defect, defect clustering, integrated circuits (IC), Hotelling 2 control chart, wafer. I. INTRODUCTION PRODUCT yield plays a critical role in determining the market competitiveness of integrated circuit (IC) manufacturers. All IC manufacturers try to increase their product yield. The wafer yield is defined as the ratio of the number of dies that pass the final specification test to the total number of tested dies on a wafer. Wafer defects significantly impact the wafer yield. Normally, more defects produce a lower wafer yield. However, if defects on a wafer exhibit clustering, the number of defects may not reflect the actual yield. Hence, the defect-clustering phenomenon must be included when defects formed in the process are monitored. Control charts are extensively adopted in relation to industrial applications. A control chart can clearly describe all the parts in a process. The conventional defect control chart employed in support of the IC manufacturing process assumes that wafer defects follow a Poisson distribution. This assumption implies that the probability of a defect being present on a wafer is the same at all locations. All defects are independent of each other. That is, the defects are randomly distributed. However, wafer manufacturing is becoming increasingly complex as the size of wafers increases, resulting in the clustering of defects on the wafers. Under such conditions, the conventional defect control chart results in many false alarms. Albin and Friedman [1] developed Manuscript received December 17, 2002; revised July 12, L.-I. Tong is with the Department of Industrial Engineering and Management, National Chiao Tung University, HsinChu, Taiwan, R.O.C. ( litong@cc.nctu.edu.tw). C.-H. Wang is with the Department of Computer Science, Chung Cheng Institute of Technology, National Defense University, Taoyuan, Taiwan, R.O.C. ( wch@ccit.edu.tw). C.-L. Huang is with the Process Engineering Department, Advanced Semiconductor Manufacturing Group, Kaohsiung, Taiwan, R.O.C. ( charliehuang@aseglobal.com). Digital Object Identifier /TSM a revised defect control chart based on the Neyman Type-A distribution of the compound Poisson distribution family to reduce the number of false alarms obtained using conventional control charts. The control limits of this revised defect control chart exceed those of conventional defect control charts. Accordingly, Albin and Friedman s control chart reduces the number of false alarms obtained using the conventional defect control chart. However, this approach does not account for the defect clustering effects, so it cannot actually reflect the defects circumstances formed in IC fabrication to trace the causes of a defect, as engineers would require. Some studies of defect clustering have been developed, such as those of Stapper [8], [9], Cunningham [2], and Jun et al. [4]. Among these studies, Jun et al. [4] developed a clustering index (CI) that could more accurately depict the clustering phenomenon than other studies. The CI value is independent of the chip area and does not require any assumptions about the defect distribution. This study therefore develops a multivariate Hotelling control chart based on the number of defects and CI values to monitor efficiently the defects in IC fabrication. The CI is utilized to quantify clustering. The simultaneous monitoring or control of several related quality characteristics is necessary in many situations. Even when these quality characteristics are independent, monitoring these variables independently could be very misleading [7]. Since the multivariate control chart is more effective for monitoring two or more related quality characteristics than employing the control chart for each variable, this study proposes a novel procedure for constructing a multivariate Hotelling control chart using defect number and CI as two quality characteristics to monitor or control defect number and defect clustering on wafers. Meanwhile, the statistics are decomposed based on Mason et al. [6] to trace the source of the out-of-control points on the multivariate Hotelling control chart. The decomposition of statistics helps to identify the determinant of the out-of-control points, whether too many defects, the clustering of defects, or both. A case study of a Taiwanese IC manufacturer demonstrates the effectiveness of the proposed procedure. II. LITERATURE REVIEW The conventional defect control chart employed in the IC manufacturing process assumes that wafer defects follow a Poisson distribution. The probability of a sample point being outside the above limits approximates when the process is under control. For details on defect control charts, refer to Montgomery [7] /$ IEEE

2 TONG et al.: MONITORING DEFECTS IN IC FABRICATION USING HOTELLING CONTROL CHART 141 Fig. 1. Defect maps and projected x and y coordinates [4]. Stapper [8] indicated that the defect distribution exhibits clustering along with increased wafer size, making the defect distribution no longer random. In this case, the conventional defect control chart is inappropriate for monitoring defects on wafers since it produces many false alarms. Some revised defect control charts have been developed to solve this problem. For example, Albin and Friedman [1] developed a revised defect control chart based on the Neyman Type-A distribution of the compound Poisson distribution family. They assumed that the clusters follow a Poisson distribution and that defects within clusters also follow a Poisson distribution. The control limits based on the Neyman assumption are times those obtained on the Poisson assumption. The approach of Albin and Friedman [1] reduces the number of false alarms by widening these control limits. However, Stapper [8] stated that this revised control chart does not account for defect clustering. Defect clustering on a wafer may produce a yield inconsistent with that obtained when defects are randomly distributed, given a particular number of defects, because the relationship between the number of defects and the degree of defect clustering was not considered in determining the yield. Consequently, defect clustering must be considered when determining the effects of defects on wafer yield. Jun et al. [4] proposed a new cluster index (CI) to evaluate the effect of the clustering of defects in a wafer. The CI value is independent of the chip area and does not depend on assumptions concerning the distribution of defects. Restated, suppose a wafer has defects. In and coordinates, the th defect can be represented as, where. Accordingly, all of the defects can be projected into and coordinates. Fig. 1 [4] presents examples of defect maps and the corresponding projected and coordinates. The defect intervals on the and coordinates are defined as follows [4]: (1) (2) where. According to Fig. 1(d), defect clustering tends to present burstiness or clumps in and coordinates. This type of defect clustering results in a large variance of defect intervals. However, Fig. 1(b) and (c) shows that burstiness on either the or the axis does not necessarily reveal clustered defects. Based on the relationship between defect clustering and cluster burstiness or clumps in and coordinates, Jun et al. [4] proposed the following CI: (3) TABLE I COMPARISONS OF AND CI where, and. If defect locations are assumed to be uniformly and randomly distributed, the index CI is close to one. A larger CI value is associated with highly clustered defects. Jun et al. [4] also compared the performance of CI with the clustering parameter obtained from the well-known negative binomial model [9]. Table I [4] presents the comparisons and reveals that the CI can more accurately quantify clustering than can, because the distribution of is quite scattered, making evaluation of the clustering of defects inconvenient. Therefore, this study uses CI and the number of defects on a wafer to establish a multivariate Hotelling control chart for monitoring the characteristics of defects. A case study of a Taiwanese IC manufacturer demonstrates the effectiveness of the proposed procedure. III. PROPOSED PROCEDURE The proposed multivariate Hotelling control chart, based on the CI index and the number of defects, is generated in the following six steps. Step 1) Obtain a wafer map using the KLA 2110 wafer inspection system. The KLA 2110 wafer inspection system is extensively used wafer inspection equipment in the semiconductor industry. This equipment performs in-line inspection during a manufacturing process. After wafers are inspected using KLA, a wafer map that includes the number of defects, their sizes, and their locations on the inspected wafer, is produced. These collected data are then utilized to monitor the wafer defects produced in IC fabrication. Step 2) Calculate the total number of defects and the CI values. The total number of defects and the CI values of the wafer can be calculated from the locations, sizes, and types of defects, obtained from the wafer map using (1) (3).

3 142 IEEE TRANSACTIONS ON SEMICONDUCTOR MANUFACTURING, VOL. 18, NO. 1, FEBRUARY 2005 Step 3) Step 4) Perform a normality test on the number of defects and CI values. A goodness of fit test is performed to determine the normality of the number of defects and CI values, obtained from Step 2. A normality transformation is performed on the number of defects and the CI values if the data violate the normality assumption. A trial-and-error process in which the practitioner simply tries one or more transformations is a widely used method for selecting the transformation for achieving normality [10]. Moreover, the analytical procedure is also an effective means to identify the type of power transformation to achieve normality [6]. Perform outlier analysis. Outliers must be detected and the corresponding causes traced to determine whether the outlier should be deleted or corrected, since outliers can severely influence the results of the analysis. This study uses the Forth Spread method [3] to detect outliers. Let and be the first and third quartiles, and ; then, the lower and upper limits of the nonoutlier data and are expressed as (4) (5) The data beyond the limits are referred to as outliers. The outliers must be deleted or corrected before the multivariate Hotelling control chart is established, if assignable causes exist. Step 5) Establish the multivariate Hotelling control chart. After outlier analysis is performed, a multivariate Hotelling control chart is established based on the number of transformed defects and the CI values considered as two quality characteristics. A multivariate Hotelling control chart can effectively monitor multiple responses, especially when a moderate or strong correlation exists among them. Suppose pieces of wafers ( subgroups), quality characteristics, and one observation are collected for each subgroup. The Hotelling statistics are determined as [11] Step 6) (6) where represents the -dimensional quality characteristics and and are the common estimators for the mean vector and covariance matrix obtained from these values of quality characteristics. The upper control limit of the multivariate Hotelling control chart can be expressed as follows [11]: where represents a significance level. Trace the sources of the out-of-control points. (7) TABLE II COORDINATES OF DEFECTS ON THE FIRST WAFER The statistics are decomposed following the method of Mason et al. [6] for exploring the sources of out of control points on a multivariate Hotelling control chart. The interactions among the responses are determined by statistical decomposition. In this study, the number of defects and CI values were the two correlated quality characteristics. Suppose that represents an out-of-control point; can be decomposed as, where and represent the effects of the number of defects and defect clustering, respectively. represents the interactions between the number of defects and the defect clustering under the condition of defect clustering and represents the interactions between the number of defects and the defect clustering under the condition of the number of defects. The individual decomposed statistic can be used to trace the source of the out-of-control point, if its statistical value exceeds the corresponding UCL as follows [6]: IV. ILLUSTRATION The effectiveness of the proposed procedure was demonstrated using a metal 2 etch process performed by a Taiwanese IC manufacturer. The collected data were analyzed as follows. Step 1) Obtain the wafer map using the KLA 2110 wafer inspection system. After metal 2 etching, 110 pieces of six-in wafer maps were obtained using KLA Each wafer included 396 chips. Table II lists the absolute coordinates of nine defects on the first wafer. The origin of the coordinate system is the bottom left of the wafer. Step 2) Calculate the total number of defects and CI values. The CI values can be obtained by substituting the coordinates of 110 wafers into (1) (3). Table III summarizes the number of defects and CI values of the 110 wafers. Step 3) Perform the normality test on the number of defects and CI values. Fig. 2 displays the distribution of CI values. In this figure, the CI value distribution is nonnormal. Since the distribution of CI values is skewed to high (8)

4 TONG et al.: MONITORING DEFECTS IN IC FABRICATION USING HOTELLING CONTROL CHART 143 Fig. 2. CI value distribution. Fig. 3. Distribution of ln (CI). Step 4) TABLE III NUMBER OF DEFECTS AND CI VALUES values, the natural log transformation of CI values produces a distribution that is closed to normal. Fig. 3 displays the corresponding distribution. Similarly, the distributions of the number of defects and the transformed number of defects are also plotted. Figs. 4 and 5 display the corresponding plots. Conduct outlier analysis. The lower and upper limits of the nonoutlier for the transformed number of defects, and can be obtained by substituting the transformed numbers of defects on the 110 wafers into (4) and (5). Similarly, and on the transformed CI values, 1.54 and 2.67 can be obtained by substituting the transformed CI values into (4) and (5). No outliers were present because all the transformed number of defects and CI values associated with the 110 wafers were located between and. Step 5) Establish multivariate Hotelling control chart. Hotelling statistics for the 110 wafers were calculated by substituting the transformed number of defects and CI values into (6). Table IV lists the corresponding calculations at a significance level of. The UCL of the multivariate Hotelling control chart was obtained from (7) with and a specified significance level. Figs. 6 and 7 present the multivariate Hotelling control charts with and and include three

5 144 IEEE TRANSACTIONS ON SEMICONDUCTOR MANUFACTURING, VOL. 18, NO. 1, FEBRUARY 2005 Fig. 4. Distribution of the number of defects. Fig. 5. Distribution of ln (number of defects). TABLE IV HOTELLING T STATISTICS FOR =0:05 TABLE V T STATISTICS FOR OUT-OF-CONTROL POINTS FOR =0:05 AND =0:10 Step 6) and nine out-of-control points, respectively. Table V lists the corresponding statistics. Trace the sources of out-of-control points. The decomposed statistics for and were calculated using the method of Mason et al. [6] to identify the sources of the out-of-control points. The corresponding critical points (that is, UCL) were calculated using (8) to test whether the individual decomposed statistics yielded points outside the control limits. Tables VI and VII present the results of the calculations.

6 TONG et al.: MONITORING DEFECTS IN IC FABRICATION USING HOTELLING CONTROL CHART 145 Fig. 6. Hotelling T control chart for =0:05. Fig. 7. Hotelling T control chart for =0:10. TABLE VI DECOMPOSITION OF T STATISTIC FOR =0:05 TABLE VII DECOMPOSITION OF T STATISTIC FOR =0:10 According to Tables VI and VII, if the interactions are the sources of the out-of-control points, then a low correlation exists between these two quality characteristics. That is, one of two possible situations pertains. The first involves many defects that are not clustered on a wafer, as for wafers 9 and 80. The second involves a few defects with severe clustering, as for wafer 102. The results of the proposed procedure were compared with those obtained using a conventional defect control chart and the modified defect control chart of Albin and Friedman [1]. Fig. 8 presents the conventional defect control chart based on 110 wafers and shows 23 out-of-control points. The mean number of defects is , indicating that the process is significantly out of control. However, these 110 wafers do not exhibit the expected low yield because the defects cluster on

7 146 IEEE TRANSACTIONS ON SEMICONDUCTOR MANUFACTURING, VOL. 18, NO. 1, FEBRUARY 2005 Fig. 8. Conventional defect control chart based on 110 wafers. Fig. 9. Revised defect control chart. wafers, causing the conventional defect control chart to produce many false alarms. Following the approach of Albin and Friedman [1], the parameters of the Neyman Type-A distribution, and are and , respectively, based on the mean and standard deviation of the number of defects on 110 wafers. The probability density function of the Neyman Type-A distribution is therefore obtained and the corresponding UCL is 180. Fig. 9 displays this control chart and reveals only one out-of-control point because Albin and Friedman [1] tried to resolve the false alarm problem in conventional defect control charts. Although Albin and Friedman s approach overcomes the shortcoming of conventional defect control charts, their study did not consider the clustering of defects effect. Therefore, Albin and Friedman s approach cannot accurately elucidate the true circumstances of defect formation or provide efficient process information to engineers. V. CONCLUSION The number and distribution of defects significantly affects wafer yield. As wafer size increases in IC design, the clustering of defects in IC fabrication is becoming more pronounced. In such circumstances, the conventional defect control chart is unsuitable because it produces many false alarms. Therefore, in addition to the number of defects, the clustering of defects must also be involved in inline process control to elucidate accurately the circumstances of the defects. This study proposed an efficient multivariate Hotelling control chart based on the number of defects and clustering index (CI). Through statistical decomposition, the source of out-of-control points can be investigated. Engineers can obtain efficient process information from the proposed multivariate control chart to solve the problem of defects. In summary, this study has the following merits. 1) The proposed multivariate control chart, which includes information on the number and clustering of defects, can efficiently monitor the wafer manufacturing process circumstances and reduce the complexity of process control. 2) This study further analyzes the interaction between the defect clustering and the number of defects using statistical decomposition. Clustering phenomena are explicitly described and the sources of out-of-control points

8 TONG et al.: MONITORING DEFECTS IN IC FABRICATION USING HOTELLING CONTROL CHART 147 are accurately determined. The information on out-ofcontrol points helps engineers to trace technically corresponding causes (such as human, machine-related, material, methodological, and others.). 3) The proposed multivariate control chart can be generalized to any size of a wafer. Lee-Ing Tong is a Professor in the Department of Industrial Engineering and Management, National Chiao-Tung University, Taiwan, R.O.C. She has published several journal articles in the areas of quality engineering and statistical process control. REFERENCES [1] S. L. Albin and D. J. Friedman, The impact of clustered defect distributions in IC fabrication, Manage. Sci., vol. 35, no. 9, pp , [2] J. A. Cunningham, The use and evaluation of yield models in integrated circuit manufacturing, IEEE Trans. Semiconduct. Manufact., vol. 3, pp , May [3] D. C. Hoaglin, B. Iglewicz, and J. W. Tukey, Performance of some resistant rules for outlier labeling, J. Amer. Statistical Assoc., vol. 81, pp , [4] C. H. Jun, Y. Hong, S. Y. Kim, K. S. Park, and H. Park, A simulationbased semiconductor chip yield model incorporating a new defect cluster index, Microelectronics Reliability, vol. 39, pp , [5] N. Johnson and D. Wichern, Applied Multivariate Statistical Analysis. Englewood Cliffs, NJ: Prentice-Hall, [6] R. L. Mason, N. D. Tracy, and J. C. Young, A practical approach for interpreting multivariate T control chart signal, J. Quality Technol., vol. 29, pp , [7] D. C. Montgomery, Introduction to Statistical Quality Control. New York: Wiley, [8] C. H. Stapper, The effects of wafer to wafer density variations on integrated circuit defect and fault density, IBM J. Res. Devel., vol. 29, pp , [9], Defect density distribution for LSI yield calculations, IEEE Trans. Electron Devices, vol. ED-20, pp , [10] S. Sharma, Applied Multivariate Techniques. New York: Wiley, 1996, pp [11] N. D. Tracy, J. C. Young, and R. L. Mason, Multivariate control chart for individual observations, J. Quality Technol., vol. 24, pp , Chung-Ho Wang is an Associate Professor in the Department of Computer Science, Chung Cheng Institute of Technology, National Defense University, Taiwan, R.O.C. He has published several journal articles in the areas of quality engineering and statistical process control. Chih-Li Huang is an Engineer in the Process Engineering Department, Advanced Semiconductor Engineering Group, Taiwan, R.O.C. His research interests include the area of statistical process control.